Let $X_n$ be a Hypergeometric random variable, with parameters $N,K,n$, ($N$ the population size, $K$ the special elements, $n$ the sample size).
The PDF is $$P(X_n=k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}},$$ and the CDF is $$F_n(x) \equiv P(X_n\leq x) = \sum_{k=0}^{\lfloor x \rfloor} \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}.$$
Assuming now that $N=2n$ and $K$ is fixed, what is the limit of the CDF function, as $n\to \infty$ ?
Correction: $K$ is not fixed, but rather $K=\Theta (n)$.